A review of Logarithms
...tem of logarithms is due to the combined effort of Napier and Henry Biggs in 1624. Natural logarithms first arose as more or less accidental variations of Napier's original logarithms. Their real significance was not recognized until later. The earliest natural logarithms occur in 1618. Logarithms are useful in many fields from finance to astronomy. Shortcuts Multiplication is a shortcut for addition. Recall that means 5 + 5+ 5. Exponents are a shortcut for multiplication. Recall that means . Logarithm is a shortcut for exponents. Function Before we review exponential and logarithmic functions, let's review the definition of a function and the graph of a function. A function is just a rule. The rule links one number to a second number in an orderly and specific manner. All the points on the graph of a function are made up of two parts: (a number, and the function value at that number). For example, the number of hours worked in a week could be the first number, and the salary for the week could be the function value. If an hourly salary is $7.00, then the rule would be 7 times the number of hours worked. Many students in high school and in college have a difficult time with logarithms. In many cases, they memorize the rules without fully understanding them, and they sometimes even manage to squeak by a course. Why waste their time on these archaic entities; they are never going to see them again. Wrong! Just when the student breathes a sigh of relief to be done with logarithms, they encounter them again in another course. They are now in trouble because the second encounter with logarithms is at a more sophisticated level. Without an understanding of the basics, the student is doomed to blindly stumble through and fail the course. You have our sympathy and you have our solution. We at S.O.S. Math want you to succeed. We have prepared a review of logarithms for you with examples and problems. You can start at the beginning or jump in at any place. Since logarithms are exponents, we will review exponential functions before we review logarithms and logarithmic functions. Before we review exponential functions, we will present a brief history of logarithms and a brief discussion of the functions. You could identify a point on the graph of a function as (x,y) or (x, f(x)). You may have only one function value for each x number. If the points (2, 3), (4, 5), (10, 11), and (25, 26) are located on the graph of a function, you could easily figure out a corresponding rule. To get the function value, you just add 1 to the first number. The rule is f(x) = x + 1. The points (3, 8) and (3, 18) could not be points on the graph of a function because there are two different function values for the same x value. Equivalence Before you choose a topic from the list, let's review what you know about the word "equivalent." You will come across this word in all your math courses. According to Webster, the word "equivalent" means to have equal value, equal amount, equal measure, equal area, equal length, etc. You get the picture. For example, 1 yard is equivalent to 3 feet because both represent the same measure. You will see the notation 1 yard = 3 feet. Obviously, the left side of the = sign is not identical to the right side. The numbers and the words are different. However, the left side is equivalent to the right side because both sides denote the same length. In this case, the = sign is used to denote equivalence. Suppose you wish to purchase an item for$1.20, you could pay for it with a dollar bill and 2 dimes, you could pay for it with 12 dimes, you could pay it for 4 quarters and 4 nickels, etc. If you put each method of payment in a separate pile, you will note that they do not look the same. They are in fact not identical. However, they are equivalent because all the methods of payment tota...